Longtime existence of the Kähler-Ricci flow on ℂⁿ

Abstract

We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on C n \mathbb {C}^n without assuming the initial metric has bounded curvature, thus extending results in an earlier work of the authors. We prove the existence of a longtime bounded curvature solution emerging from any complete U ( n ) U(n) -invariant Kähler metric with non-negative holomorphic bisectional curvature, and that the solution converges as t → ∞ t\to \infty to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general Kähler metrics on C n \mathbb {C}^n which are not necessarily U ( n ) U(n) -invariant.